p(x)-Harmonic functions with unbounded exponent in a subdomain

• Author(s): Manfredi, Juan J. ; Rossi, Julio D. ; Urbano, José Miguel
• Date: 2008
• Persistent ID: http://hdl.handle.net/10316/11222
• Origin: Estudo Geral - Universidade de Coimbra
• Subject(s): p(x)-Laplacian; Infinity-Laplacian; Viscosity solutions
• Document Type: Preprint
• Language: English

Description

We study the Dirichlet problem −div( ∇u p(x)−2∇u) = 0 in , with u = f on @ and p(x) = ∞ in D, a subdomain of the reference domain . The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as n → ∞ of the solutions un to the corresponding problem when pn(x) = p(x)∧ n, in particular, with p = n in D. Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem. Moreover, we examine this limit in the viscosity sense and find an equation it satisfies.